Similarity to Physical Coordinate Transform

The Falkner-Skan and Falkner-Skan-Cooke solvers return profiles on \(\eta\). This transform maps those profiles to physical \(y\).

Conventions

The similarity solutions use the nondimensional temperature profile

\[ \tau(\eta) = \frac{T(\eta)}{T_e} \]

where \(T_e\) is the boundary-layer edge temperature at the streamwise station of interest. For a perfect gas boundary layer with \(\partial p/\partial y = 0\), the density relation is

\[ \rho = \frac{\rho_e}{\tau} \]

(For the derivation see Levy-Lees and Illingworth-Stewartson.)

The edge velocity is represented by the Falkner-Skan power law

\[ u_e(x) = Cx^m \]

and the Hartree pressure-gradient parameter is

\[ \beta_H = \frac{2m}{m+1} \]

General Inverse Form

Both implemented transforms are density-weighted wall-normal coordinates. After differentiating the chosen definition of \(\eta\) with respect to \(y\), the density factor is replaced using \(\rho = \rho_e/\tau\). The remaining factors depend only on the local station and the selected transform, so they are collected into a single inverse length scale \(\eta_s\).

With that notation, both implemented transforms can be written locally as

\[ \frac{\partial \eta}{\partial y} = \frac{\eta_s}{\tau} \]

The scale \(\eta_s\) is therefore not an additional assumption. It is just the coefficient multiplying \(1/\tau\) after the coordinate transform has been differentiated. Its exact definition is different for the Levy-Lees and Illingworth-Stewartson transforms.

Levy-Lees: η definition and η_s

\[ \eta = \sqrt{\frac{(m+1)\rho_e u_e}{2\mu_e x}}\int_0^y \frac{\rho}{\rho_e}\,dy' \]

Differentiating with respect to \(y\) and substituting \(\rho/\rho_e = 1/\tau\):

\[ \frac{\partial\eta}{\partial y} = \frac{1}{\tau}\underbrace{\sqrt{\frac{(m+1)\rho_e u_e}{2\mu_e x}}}_{\eta_{s,LL}} \]

Illingworth-Stewartson: η definition and η_s

\[ \eta = \sqrt{\frac{(m+1)u_e}{2\nu_{ref}x}}\int_0^y \frac{\rho}{\rho_e}\,dy' \]

where \(\nu_{ref} = \mu_{ref}/\rho_{ref}\) is a prescribed reference kinematic viscosity (defaults to the local edge value \(\nu_e = \mu_e/\rho_e\)).

Differentiating with respect to \(y\) and substituting \(\rho/\rho_e = 1/\tau\):

\[ \frac{\partial\eta}{\partial y} = \frac{1}{\tau}\underbrace{\sqrt{\frac{(m+1)u_e}{2\nu_{ref}x}}}_{\eta_{s,IS}} \]

With the default reference state (\(\nu_{ref} = \nu_e\)), \(\eta_{s,IS} = \eta_{s,LL}\).

Inverting gives

\[ \frac{\partial y}{\partial \eta} = \frac{\tau}{\eta_s} \]

Integrating from the wall, where \(\eta = 0\) and \(y = 0\),

\[ y(\eta) = \frac{1}{\eta_s}\int_0^\eta \tau(\hat{\eta})\,d\hat{\eta} \]

Thus the transform has two pieces:

Compute the scale \(\eta_s\) for the chosen similarity transformation.
Integrate \(\tau(\eta)\) over the similarity grid.

Available Transforms

Transform	Use
Levy-Lees	Default inverse map for the `falkner_skan` equation family
Illingworth-Stewartson	Default inverse map for the `falkner_skan_cooke` equation family